Two similar right triangles have areas of 6 square inches and 150 square inches. The length of the hypotenuse of the smaller triangle is 5 inches. What is the sum of the lengths of the legs of the larger triangle?
Answer: Since the smaller triangle has hypotenuse 5, we guess that it is a 3-4-5 triangle.  Sure enough, the area of a right triangle with legs of lengths 3 and 4 is $(3)(4)/2 = 6$, so this works.  The area of the larger triangle is $150/6=25$ times the area of the smaller triangle, so its side lengths are $\sqrt{25} = 5$ times as long as the side lengths of the smaller triangle.  Therefore, the sum of the lengths of the legs of the larger triangle is $5(3+4) = \boxed{35}$.

Proof that the only possibility for the smaller triangle is that it is a 3-4-5 triangle: Let's call the legs of the smaller triangle $a$ and $b$ (with $b$ being the longer leg) and the hypotenuse of the smaller triangle $c$. Similarly, let's call the corresponding legs of the larger triangle $A$ and $B$ and the hypotenuse of the larger triangle $C$. Since the area of the smaller triangle is 6 square inches, we can say  $$\frac{1}{2}ab=6.$$ Additionally, we are told that the hypotenuse of the smaller triangle is 5 inches, so $c=5$ and $$a^2+b^2=25.$$ Because $\frac{1}{2}ab=6$, we get $ab=12$ or $a=\frac{12}{b}$. We can now write the equation in terms of $b$. We get \begin{align*}
a^2+b^2&=25\\
\left(\frac{12}{b}\right)^{2}+b^2&=25\\
12^2+b^4&=25b^2\\
b^4-25b^2+144&=0.
\end{align*} Solving for $b$, we get  $$b^4-25b^2+144=(b-4)(b+4)(b-3)(b+3)=0.$$ Since we said that $b$ is the longer of the two legs, $b=4$ and $a=3$. Therefore, the triangle must be a 3-4-5 right triangle.